Type Classes in Haskell Tom Schrijvers. Leuven Haskell User Group

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1 Type Classes in Haskell Tom Schrijvers Leuven Haskell User Group

2 Haskell Research Team Partners Monads Type Classes GHC Folds Pattern Matching Equational Reasoning DSLs Advanced Types

3 Adhoc Overloading

4 Example: Addition Different Implementations for different types Int -> Int -> Int Float -> Float -> Float Vector -> Vector -> Vector How do we name the implementations?

5 Different Names for Different Implementations e.g. Ocaml + : int -> int -> int +. : float -> float -> float! remember more names common concept / properties

6 Same Name for Different Implementations e.g. Java + : int -> int -> int + : float -> float -> float adhoc overloading operator overloading! hard-wired in the language not user-extensible adhoc polymorphism

7 Type Classes for Adhoc Overloading

8 Type Classes How to make ad-hoc polymorphism less ad hoc Philip Wadler and Stephen Blott

9 Example: Equality (==) :: a -> a -> Bool equality is adhoc overloaded > "hello" == "world" False > 2 == (1+1) True

10 Example: Equality (==) :: a -> a -> Bool not every type supports equality > id == (\x -> x) No instance for (Eq (a -> a)) arising from a use of == In the expression: id == (\ x -> x)

11 Constraint Polymorphism qualified type type class constraint type class (==) :: Eq a => a -> a -> Bool type a must have an implementation of equality

12 Type Class Declaration class Eq a where (==) :: a -> a -> Bool method

13 Type Class Instantiation data Color = Red Blue instance Eq Color where Red == Red = True Blue == Blue = True _ == _ = False > Red == Red True > Red == Blue False

14 Using Methods isred :: Color -> Bool isred c = c == Red ok because Eq Color instance exists

15 Constraint Polymorphic Use most general type? same :: Eq a => (a,a) -> Bool same (x,y) = x == y allsame :: Eq a => [(a,a)] -> Bool allsame l = all same l type class constraints propagate!

16 Multiple Methods class Eq a where (==) :: a -> a -> Bool (/=) :: a -> a -> Bool

17 Type Class Instantiation data Color = Red Blue instance Eq Color where Red == Red = True Blue == Blue = True _ == _ = False x /= y = not (x == y)

18 Default Implementations class Eq a where (==) :: a -> a -> Bool (/=) :: a -> a -> Bool x /= y = not (x == y)

19 Default Implementation data Color = Red Blue instance Eq Color where Red == Red = True Blue == Blue = True _ == _ = False x /= y = not (x == y)

20 Default Implementations class Eq a where (==) :: a -> a -> Bool x == y = not (x /= y) (/=) :: a -> a -> Bool x /= y = not (x == y)

21 Instance Contexts instance context instance Eq a => Eq [a] where [] == [] = True (x:xs) == (y:ys) = x==y && xs==ys _ == _ = False

22 Instance Contexts instance (Eq a,eq b) => Eq (a,b) where (x1,x2) == (y1,y2) = x1 == y1 && x2 == y2

23 Subclassing super class class Eq a => Ord a where (<=) :: a -> a -> Bool every instance of Ord must also have an instance of Eq r :: Ord a => a -> a -> Bool r x y = x == y

24 The Num Class class Num a where (+) :: a -> a -> a (-) :: a -> a -> a (*) :: a -> a -> a negate :: a -> a abs :: a -> a signum :: a -> a frominteger :: Integer -> a

25 Dictionary-Passing Translation

26 Implementation Source Haskell with type classes translation in the GHC type checker Target Haskell without type classes GHC Core What does the target look like? How is the target obtained?

27 Dictionary Representation class Eq a where (==) :: a -> a -> Bool (/=) :: a -> a -> Bool data Eq a = Eq { (==) :: a -> a -> Bool, (/=) :: a -> a -> Bool } (function) dictionary

28 Instance Translation instance Eq () where (==) = \x y -> True (/=) = \x y -> False equnit :: Eq () equnit = Eq { (==) = \x y -> True, (/=) = \x y -> False }

29 Call Translation p :: () -> () -> Bool p x y = x == y p :: () -> () -> Bool p x y = (==) equnit x y typically followed by inlining

30 Instance Translation instance (Eq a,eq b) => Eq (a,b) where (==) = \(x1,y1) (x2,y2) -> x1 == x2 && y1 == y2 eqpair :: Eq a -> Eq b -> Eq (a,b) eqpair da db = Eq { (==) = \(x1,y1) (x2,y2) -> (==) d1 x1 x2 && (==) d2 y1 y2, }

31 Call Translation q :: ((),()) -> ((),()) -> Bool q x y = x == y p :: ((),()) -> ((),()) -> Bool p x y = (==) d x y where d = eqpair equnit equnit

32 Constraint Polymorphic Signature Translation r :: Eq a => a -> a -> Bool r x y = x == y r :: Eq a -> a -> a -> Bool r d x y = (==) d x y

33 Constraint Polymorphic Signature Translation s :: Eq a => (a,a) -> (a,a) -> Bool s x y = x == y s :: Eq a -> (a,a) -> (a,a) -> Bool s d x y = (==) d x y where d = eqpair d d

34 Super Class Translation class Eq a => Ord a where (==) :: a -> a -> Bool (/=) :: a -> a -> Bool data Ord a = Ord { super :: Eq a, }

35 Super Class Access r :: Ord a => a -> a -> Bool r x y = x == y r :: Ord a -> a -> a -> Bool r d x y = (==) d x y where d = super d

36 Resolution

37 Resolution Process [(True, a )] == [(False, b )] (==)? [] [(True, a )] [(False, b )] Eq [(Bool,Char)]

38 Resolution Process Eq [(Bool,Char)] eqlist Eq (Bool,Char) eqpair eqbool Eq Bool Eq Char eqchar instance Eq a => Eq [a] instance (Eq a, Eq b) => Eq (a,b) instance instance Eq Bool Eq Char

39 Resolution Process (==) d [] [(True, a )] [(False, b )] Eq [(Bool,Char)] eqlist Eq (Bool,Char) eqpair eqbool Eq Bool Eq Char eqchar where d = eqlist (eqpair eqbool eqchar)

40 Summary

41 Type Classes Systematic approach to adhoc overloading implemented by means of dictionary passing where resolution assembles the dictionaries.

42 LANGUAGE Pragmas MultiParamTypeClasses FunctionalDependencies FlexibleContexts FlexibleInstances UndecidableInstances OverlappingInstances IncoherentInstances non-terminating resolution possible incoherent resolution possible incoherent resolution likely

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